mining supply chain optimization under geological uncertainty

8/12/2019 Mining Supply Chain Optimization Under Geological Uncertainty

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Les Cahiers du GERAD ISSN: 07112440

Mining Supply Chain Optimization

under Geological Uncertainty

R. Goodfellow

R. Dimitrakopoulos

G201354

September 2013

Les textes publies dans la serie des rapports de recherche HEC nengagent que la responsabilite de leurs auteurs. La publication de

ces rapports de recherche beneficie dune subvention du Fonds de recherche du Quebec Nature et technologies.


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Mining Supply Chain Optimization

under Geological Uncertainty

Ryan Goodfellow

Roussos Dimitrakopoulos

COSMO Stochastic Mine Planning LaboratoryDepartment of Mining and Materials Engineering

McGill University

Montreal (Quebec) Canada, H3A [email protected]

[email protected]

and GERAD

September 2013

Les Cahiers du GERAD

G201354

Copyright c 2013 GERAD


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ii G201354 Les Cahiers du GERAD

Abstract: Mining operations can be modelled as a supply chain from the sources of raw material (mines)through various processing streams to the final saleable products sold to customers. Existing research inmine design optimization has predominantly focused on the extraction sequence of materials and neglectsthe impact that selecting the optimal processing paths has on the economic viability of a mining operation.

Additionally, the vast majority of existing models assume that the value of the extracted material must becalculated for each block individually, and neglects the opportunity to blend material together to prolongthe life of the resource and generate higher cash flows. One major contributing factor for not modellingsuch complexities is the difficulty in modelling and optimizing with non-linear relationships that occur in thesupply chains processing streams.

This paper addresses the issue of selecting the optimal block destinations and processing streams through-out a mining supply chain for complex blending operations. This non-linear optimization formulation is solvedusing a particle swarm optimization algorithm. The k-means clustering technique is used to aggregate theprocessing decision variables and substantially reduce the size of the problem, leading to more efficient solu-tion times with minimal loss in the quality of the resultant solution. By clustering the decision variables andapplying the same decisions over a set of geological simulations, the proposed method also provides complexdestination policies for material extracted from a mine under geological uncertainty, somewhat akin to a moreadvanced cut-off policy. This method is tested at Vales Onca Puma nickel laterite deposit, located in Para,Brazil.

Key Words: Supply chain optimization; extractive blending; clustering; process streams; geological uncer-tainty.

Acknowledgments: The work in this paper was funded from NSERC CDR Grant 335696 and BHP Billiton,NSERC Discovery grant 239019 and the members of the COSMO Stochastic Mine Planning Laboratory:AngloGold Ashanti, Barrick, BHP Billiton, De Beers, Newmont, and Vale. The authors would like to thankJean-Yves Cloutier and the Mining Engineering Department at the Onca Puma mine for their help, adviceand insight related to the case study.


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Les Cahiers du GERAD G201354 1

1 Introduction

As the worlds mineral resources are continuously being depleted, mining companies are seeking to profitablyextract and process increasingly complex material. Through tremendous technological and operational im-provements, mining companies can often profitably produce products from material that was once considered

waste. This is often a result of blending material from various sources together to obtain a more homogenousand predictable material that meets a set of specifications prior to processing. As the options for sendingmaterials through different processing streams increases, a supply chain is formed, where one needs to makeoptimal decisions on where to send material extracted from the ground through to final products that aresold on the market (Sandeman et al., 2010).

Conventional mine optimization methods are often myopic in their abilities to harness the power ofblending material and processing streams within a supply chain. More often than not, these methods requirethe assumption that each block (a discretized unit of material) in an orebody has an economic value, andare independent of other blocks (Lerchs and Grossmann, 1965; Gershon, 1983; Caccetta and Hill, 2003;Ramazan and Dimitrakopoulos, 2004; Gleixner, 2008; Boland et al., 2009; Bienstock and Zuckerberg, 2010).To alleviate this issue, sets of constraints are placed in the mathematical formulations to provide limits on theincoming feed to the various destinations. These constraints can sometimes be misleading, in particular for

grade blending constraints in stochastic integer programming formulations (Ramazan and Dimitrakopoulos,2012), where the penalty costs associated with the constraint require the assumption of constant tonnageto ensure that the constraint is linear. Additionally, these conventional optimization models often assume aone-and-done approach to the processing streams, where the blocks are sent to a single destination, whichis then transformed into a saleable product. Naturally, these assumptions can be over-simplified and limitingfor complex mining operations. In more complex scenarios, it is possible to include stockpiles (Caccettaand Hill, 2003; Ramazan and Dimitrakopoulos, 2012), however these often require restrictive assumptions inorder to have a linear model of the system.

The concept of blending optimization for mineral resource supply chains has been a topic of researchover the past few decades, particularly in the coal and cement industries. Sarker and Gunn (1997) proposea successive linear programming approach tailored to solve non-linear coal-blending formulations, whereportions of the non-linear programs sub-problems are fixed and solved successively using linear programminguntil convergence is obtained. Asad (2010) proposes a linear programming-based optimization model forshort-range production in the cement industry. There have also been some applications in the metal miningindustry, albeit specifics are rarely discussed for intellectual property reasons. Hoerger et al. (19991) andHoerger et al. (19992) give an overview of their mixed integer linear programming (MILP) formulation forscheduling, stockpiling and processing for Newmonts large mining complex in Nevada. Wharton (2007) andWhittle (2007) show in several case studies the added value in blending and improving the performance ofthe supply chain, including the ability to identify bottlenecks. Stone et al. (2007) discuss an application ofBHP-Billitons in-house software (Blasor) to their Yandi deposit, an iron ore blending operation in WesternAustralia. Chanda (2007) formulates the blending and supply chain as a linear programming network wherethe objective is to minimize the costs of material flow through the network, and tests the method on ahypothetical supply chain with six mines, five concentrators, three smelters and two refineries. This methodis, however, limited in ability to model intricacies in the supply chain.

More recently, Sandeman et al. (2010) present two blending and supply chain case studies solved usingdiscrete event simulation (DES) and a linear optimization model. In the DES model, each physical item isconsidered as a discrete entity that has unique characteristics (Sandeman et al., 2010). While specifics arenot discussed, the methodology is certainly one of the most advanced works with mining applications to datebecause it is able to efficiently analyze what-if scenarios for a seemingly infinite number of possibilities (e.g.stochastic processing times and delays) at each step in the supply chain.

Other researchers attempt to optimize non-linear blending problems through the use of metaheuristics.In particular, particle swarm optimization (PSO) (Kennedy and Eberhart, 1995) appears to be an attractivemethod because of its ability to solve both discrete, continuous and mixed (discrete with continuous) non-linear optimization problems, which permits a high degree of flexibility for modelling complex supply chains.


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2 G201354 Les Cahiers du GERAD

Li et al. (2011) apply a hybridized particle swarm method with genetic algorithms to a cement blendingoperation, where the genetic mutations are used to ensure population diversity as the solution converges.Yanbin (2008) apply a hybridized simulated annealing (Kirkpatrick et al., 1983; Geman and Geman, 1984)and particle swarm optimization algorithm to power coal blending. Both approaches show adequate results,but are limited in their applicability to model advanced cases with complex supply chains, i.e. beyond the

first stockpile or plant.

The purpose of this paper is to introduce a generalized mine supply chain optimization methodology, whichis optimized using a particle swarm optimization algorithm. First, a property-based modelling approach isdiscussed. Given the large size of the problem, a method using the k-means clustering algorithm (Lloyd, 1982)to aggregate initial block destination decisions is discussed. This concept is then extended to mining supplychain optimization under geological uncertainty, where the clustering is performed over a set of simulations,leading to more robust long-term decision making. A particle swarm optimization algorithm for optimizingthe proposed mining supply chain models is then discussed. The method is then applied to Vales Onca Pumamine, a nickel laterite deposit in Para State, Brazil. Finally, conclusions and future work are presented.

2 Optimization of Mining Supply Chains with Blending

For the purpose of this approach, it is assumed that the production schedule for the mine(s) is static. Theoptimization method does not attempt to generate a new production schedule for each of the mines; thisis a topic for future research. The goal is to develop a generalized optimization framework that can beapplied to many different types of mining complexes and can be solved efficiently. In order to generalizethis framework, an approach to modelling mining supply chains based on materials (and their respectiveproperties) is developed. A generalized mixed-integer non-linear programming formulation can then bedeveloped and tailored to solve a specific problem. Given the fact that evaluating the mining supply chainis by far the most computationally costly function during the optimization process, an aggregation of theinitial block destinations is proposed using the k-means clustering algorithm; the aggregation is based on thegeological properties of the model, and the optimization framework can be modified to solve the clustereddestination problem. Since clustering can be performed over a set of geological simulations, rather than

each simulation uniquely, it is possible to jointly optimize all simulations, which are tied together via theinitial clustering; this is somewhat akin to a two-stage stochastic programming problem where the first-stagedecision variables choose the destinations for the clusters and the recourse variables optimize the variousprocessing streams based on the first-stage decisions made. Given that these types of problems are incrediblydifficult to solve for large-scale problems, an efficient framework using particle swarm optimization withheuristics (Kennedy and Eberhart, 1995) is proposed.

2.1 Problem modelling methodology

2.1.1 Background and definitions

Consider a set of locations in the mining supply chain, L, which can be split into two subsets: Lm L, whichrepresents the set of mines, and Ld L, which represents the set of destinations. Note that Lm Ld= L and

Lm Ld= . Let lm Lm represent a mine from the set of mines, and ld Ld represent a destination fromthe set of destinations. Let l L represent any location in the mining supply chain. Each mine lm Lmcan be discretized into blocks (or aggregates thereof), which will be referred to as Blm , where each block inthe set can be denoted as blmk Blm where k {1, . . . , N lm}is used to index each block and Nlm is the totalnumber of blocks in Blm .

The life of the entire supply chain can be represented as P ={1, . . . , P max}, where each p Prepresentsa discrete unit of time (e.g. a production period), and Pmax is the index of the maximum operating periodfor the supply chain. Each period that locationl L operates is defined as pl Pl P, where the operatingperiods for location l, pl, may or may not be contiguous. It is assumed that each block b

lmk must have an

assigned extraction period, denoted by plmk Plm , and this period is not chosen dynamically during the


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optimization (i.e. fixed block extraction periods). Integrating production scheduling with the mining supplychain is a topic of future research.

Any destination l L has a set of input materials, Il, a set of transitional materials, Tl, and a set ofoutput materials, Ol, wherebyil Il, tl Tl andol Ol are used to denote a material from their respective

sets. The mines can only have output materials, and do not receive materials for processing, i.e. for anymine lm Lm, Ilm Tlm = . For simplicity, let the agglomerated set of materials for l be represented asMl = Il Tl Ol. Let O denote the unique set of all output materials, i.e. O=

lLOl, where each output

material can be denoted by o O. It is not necessary that any ol Ol is unique when comparing to anymaterial from another destination, ol Ol , l L\l. It is noted that

lLOl may or may not be a null

set. For cases where

lLOl = , this indicates that there are multiple locations that produce the sameoutput material. For example, two mines may produce output material that would be categorized as oxides,or several refineries may produce gold bars.

Consider a directed graph G (L, OlL), where the nodesL represent the various locations, and OlL are thedirected edges that represent output material from location l L and can be used as input material to a setof candidate destinations, L Ld\l, i.e. output material (ol Ol) = (il Il) l L. In the case whereall cycles in G (L, OlL) can be removed, G (L, OlL) can form a directed acyclic graph using a topological

sorting algorithm (Cormen et al., 2009). The sorted graphGTS

(L, OlL) now represents the flow of materialsthrough the mining supply chain, and can be used as an ordering for computing updates when material flowis active throughout the supply chain. The case whereG (L, OlL) contains cycles between the vertices is notconsidered in this methodology.

Each material m Ml for location l L has a set of properties, denoted herein as Rml . The individualproperties can be denoted by rm Rml . Using the sorted graph G

TS(L, OlL), consider a destination l Land the set of destinations L L\l where (il Il) = (ol Ol) l

L, i.e. the destinations that producematerial il as an output. The input propertyrm Rmlm Il has a value v

prm

for period p, where vprm isassumed to be a linear addition of the incoming output properties, i.e.:

vprm =lL

mlOl

x

rml

vp

rml

(1)

wherex

rml

= 1 if the property rpmhas been specified to add property rml orx

rml

= 0 otherwise. Someexamples of properties that are linearly additive and can be used as input properties are total tonnages, metaltonnes, throughput hours, etc.; metal grade (%) is not a linearly additive property, hence is not acceptable.While initially this may appear confusing, the fundamental concept is simple: a property for an input materialis a linear combination of a set of property values for the various output materials from the material sources.One example might be a mill adding the incoming tonnages from three different mines; two mines mayfeed only sulfides, and the other mine may feed a small amount of oxides and sulfides combined. Using twoproperties, it is possible to count the incoming tonnages for sulfides (from three mines) separately from oxides(from one mine), and treat them differently in the processing streams.

Any intermediate or output material properties, rm Rmlm Tl Ol and l L, are functions ofpreviously calculated properties from destination l, i.e. rm=f(R

) where R is the set of all properties for

all m Ml. One common example might be a calculation of the grade, which is a non-linear function wherethe input metal tonnage property is divided by the total input tonnage property.

Given that the properties rm Rml have an ordering, it is possible to construct a directed graphG (L, OlL) for each destination l Ld, where the vertices, Rl, are all properties that must be evaluated, i.e.Rl=

mMl

Rml , and the directed arcs, Ar1r2 , are used to describe the fact that some property r1 Rlmustbe evaluated prior to evaluating property r2 Rl\r1. The property graph, Gl(Rl, Ar1r2), is assumed to notcontain cycles, and can therefore be sorted using a topological sort algorithm to yield GTSl (Rl, Ar1r2). Thisgraph can now provide the order of evaluation for each destination l Ld for when new incoming materialto the destination is added, through to the properties of the output materials.


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2.1.2 Example

To solidify this concept, consider a simple copper mine where the mined blocks go to a concentrator thenrefinery; this ordering from mine to refinery defines the sorted material graph, GTS(L, OlL). The concentra-tors incoming properties might be the incoming copper metal (tonnes) and total tonnage. An intermediate

(non-linear) property for copper grade can be generated, which is simply the tonnage of copper divided bythe total tonnes processed. This intermediate copper grade can then be used to evaluate a non-linear recoveryfunction, which can subsequently be applied to the copper tonnage to yield an output property, the recoveredcopper tonnage. Another transformed output property may be, for example, the total concentrate tonnage.These two properties can then be sent to the refinery, where the same sorts of transformations are applieduntil a final, refined, copper metal is produced and sold.

2.1.3 Algorithmic evaluation of supply chain

With the two graphs modelled, GTS(L, OlL) and GTSl (Rl, Ar1r2) l Ld, it is possible to now define the

flow of materials from the mine through the supply chain into its final product form. Two sets of decision

variables are used to define this flow. First, let the set D

blmk

denote the set of candidate destinations for

a blockblmk , and ND blmk = dD(ml)1 define the number of candidate destinations for the block. Theset D

blmk

must take into account the extraction period of the block, plmk Plm hence the availability of

the candidate destinations in that period, i.e. for each destination d D

blmk

Ld, p

lmk Pd = . Let

zd

blmk

{0, 1} d D

blmk

represent the binary decision variable of whether or not to send block blmk to

destination d, where zd

blmk

= 1 if blmk is sent to d or 0 otherwise. Note that

dD(blmk )

zd

blmk

= 1 to

ensure that the block is only (and entirely) sent to a single destination.

Next, define D (ml) as the set of candidate destinations for output material ml Ol at location l Ld.Let ytd(ml) [0, 1] define the proportion of output material ml sent to destination d D (ml) in period t.At this point, it is necessary to define whether or not the location l Ld is able to carry-over material to the

next period, or it must all be sent out in a single period. One example of a destination that may carry-overmaterial from one period to the next is a long-term stockpile that feeds a concentrator slowly over time. If thedestinationl is permitted to carry over material, it is assumed that

dD(ml)

ytd(ml) 1 ml Ol, otherwise

it must be guaranteed that all of the material leaves the process, i.e.

dD(ml)ytd(ml) = 1ml Ol.

For any given solution of variables, zd

blmk

and ytd(ml), it is now possible to evaluate the mining supply

chain as follows:

1. For each blockblmk , send (update) the linearly additive properties to its destination,d, wherezd

blmk

=

1. This is an incremental update per-block to Eq. (1).

2. For each destinationd that received material in the previous step, evaluate the subsequent propertiesin the sorted property graph, GTSd (Rd, Ar1r2).

3. For each destination d evaluated in the previous step and each period t Td (from lowest to highest),sendytd(md) of output material md Od to the subsequent destinationd D (md) in periodt

, whereD (md) is the set of destinations that accept material md as input, i.e. md Id. Since all outgoingproperties rpml md Od, p Pd associated with md are assumed to be linearly additive, the amountof this property supplied to d can be multiplied by the proportion ytd(md). It is noted that t

can beused as a lag time to control the delay between processes (e.g. long-term stockpiles).

4. For each destination d evaluated in the previous step that permits carrying over material and eachperiodt, the remaining material must be carried over to the subsequent production period subsequentperiod,t. i.e.:

vt

rml+ =

1

dD(md)

ytd(md)

vtrmd

(md) md Od, t Pd, t Pd,t < t


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5. If all destination vertices d GTS(L, OlL) have been evaluated or there does not exist a vertexd GTS(L, OlL) where material needs to be sent to a subsequent destination d D (md), quit.Otherwise, set d = d (where d is defined in Step 3), and go to Step 2.

2.2 Supply chain optimization with particle swarm optimization

2.2.1 Optimization formulation

With the previously outlined modelling methodology, it is possible to perform optimization over the min-ing supply chain. While the proposed formulation is complex to describe mathematically, the fundamentalconcept behind the formulation is simple and flexible. Given that each process can have input, intermediateand output properties, the objective is to minimize some weighted addition of the property values alongwith minimizing the weighted deviations from the set of constraints (if desired). This might be as simple asmaximizing the net present value of the metal produced from a process plant (negative weight in the objec-tive function) minus penalties for deviating from set production capacities (positive weight in the objectivefunction). Consider the following formulation:

Objective:

minlLd

tPl

mMl

rmRm

ctrm vtrmp(vrm) +dtrm atrmp(arm) +dtrm atrm

p(arm) (2)

Subject to:

vtrm atrm

Ptrm l Ld, m Ml, rm Rm, t Tl (3)

vtrm + atrm

Ptrm l Ld, m Ml, rm Rm, t Tl (4)

atrm , atrm

0 l Ld, m Ml, rm Rm, t Tl (5)

where:

vtrm is the value of property rm in period t, where rm Rml , m Ml and l Ld.

atrm is positive deviation value to ensure that constraints (3) and (4) are satisfied.

ctrm = j(rm)

(1+d(rm))t is a cost associated with property rm in period t. j(rm) is a constant value (e.g. 1 or

-1) and d (rm) is a discount rate expressed as a decimal. One common application of the value ctrm

iswhen maximizing the net present value, where j (rm) = 1 (recall that this is a minimization problem)andd (rm) would be the discount rate associated with the time value of money.

dtrm = j(devtrm)

(1+d(devtrm))t 0 and dtrm =

jdevtrm

1+ddevtrm

t 0 are penalty costs associated with the deviation

variables, atrm

andatrm

. j devtrm andj devtrm are constant penalty costs that are used to weight thedeviations (some constraints are more important to adhere to than others). d

devtrm

andd

devtrm

are

discount rates that can be used if necessary to relax the requirement on constraint deviations over time(e.g. geological risk discounting).

p (vrm),p (arm) andp

arm

are exponents associated with the property valuevrmor deviation variables,

arm and arm , respectively, and can be used to heavily penalize high deviations and only slightly forlower deviations.

Note that any propertyrmmentioned in the objective function is actually a linear or non-linear function of

the decision variableszd

blmk

andytd(ml), and that it is necessary to re-evaluate equations (2)(5) each time

the decision variables have been updated. For penalty costs dtrm and dtrm

greater than zero, the deviations


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from the constraints can be calculated using equations (6) and (7) given below, and the constraints can beremoved entirely from the formulation.

atrm = max

0, vtrm Ptrm

l Ld, m Ml, rm Rm, t Tl (6)

a

t

rm = max0, Ptrm vtrm l Ld, m Ml, rm Rm, t Tl (7)2.2.2 Solution encoding scheme

Consider a solution vector, Sol, of size N+M, where N =

lmLmNlm , M =

ldLd

tTld

mldOld

dD(mld)1,Nlm is the number of blocks in mine lm Lm,ld Ldis a destination in the set of destinations

Ld, Tld is the set of active periods for destination ld, Old is the set of output materials for ld and D (mld)is the set of destinations that accepts material mld . In other words, the first N elements ofS ol are used to

represent the block destination solution zd

blmk

, and the remaining elements are y td(mld), the proportions

of output material mld going to destination d from destination ld in period t.

Each of the elements solj j {1, . . . , N } are encoded based on the number of candidate destinations,

where the values ofsolj range from 0 to ND blmk , the maximum number of candidate destinations for

block blmk . A decoding scheme is applied to convert the value from solj

0, . . . , N

D

blmk

to the

blocks destination d where zd

blmk

= 1. The specifics of this encoding and decoding scheme are trivial and

therefore not discussed. This automatically ensures that there is only a single destination for each block,and that the block is only sent to the destination once. The remaining elements ofsolj j {N+ 1, . . . , M }must strictly adhere to the constraint that

dD(ml)

ytd(ml) 1 ml Ol for destinations that carry over

material from one period to the next, or

dD(ml)ytd(ml) = 1 ml Ol for processes that do not. The

correctness of the solution can be ensured by adding all solj j {N+ 1, . . . , M } related to destinationl Ld and output material mld Old in period t. In the case where

dD(ml)

ytd(ml) > 1 ml Ol, thevariables, solj j {N+ 1, . . . , M }, can be corrected by dividing by the total proportion of ml sent fromdestinationl in period t,dD(ml)ytd(ml).2.2.3 Aggregation of initial destination decisions using k-means clustering

For realistic mining supply chains, it is often true that the number of block destination decision variables, N,is substantially larger than the number of processing decision variables, M, whereNcan be in the millions tohundreds of millions. Given that the vast majority of the time spent is evaluating the flow of material fromthe mines to the initial destination in the supply chain (Section 2.1.3), it is of interest to reduce the amountof block destination decision variables. Some methods have been proposed in the literature to aggregateand disaggregate processing decision variables in the mine production scheduling problem (Gleixner, 2008,Boland et al., 2009). These methods rely on duality theory and linear formulations of the problem, andare therefore not necessarily the best option for mine supply chain optimization, which often requires non-linear functions and complex constraints on the blended material. One popular method for aggregating

information is k-means clustering (Lloyd, 1982), whereby groups of information can be identified based onthe similarity of their basic properties. Intuitively, aggregating processing decisions reduces the resolution onwhich decision-making can be performed, and therefore would yield lower value than a solution based solelyon block destinations. It has been experimentally found that this, however, is not the case, and that with alarger number of clusters, optimization takes much longer without significant improvements in the value ofthe objective function (Eq. (2)).

Consider the subset of blocks Bml Bl for any mine l Lm in the mining supply chain that share acommon material type m Ol. All blocks can be aggregated into n

lmclusters clusters, where each cluster is

assigned a multi-dimensional position (location) based on the average value of the properties for the blocksthat are closest to the cluster. Consider the set of properties Rm Rml , where Rml is the set of blockproperties for material m Ol at mine l Lm (e.g. grades, tonnages, or economic value). Each property


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Consider the set of destinations Linit L that can be fed by a mine, i.e. linit Linit there exists somecluster whereztlinit

cluslmk

= 1. Let Lt

oppL\Ltinitbe the set of destinations further along the supply chain

that are not fed directly by mines, where Ltopp

Ltinit

= L. The objective function in Eq. (2) can now bemodified to represent a two-stage mixed integer non-linear stochastic programming problem, as shown next.

Objective:

min

lLinit

tPl

mMl

rmRm

E

ctrm

vtrmp(vrm)

First stage decisions

+ 1

M

lLinit

tPl

mMl

rmRm

sS

dtrm atsrmp(arm ) +dtrm atsrm

p(arm)

Recourse 1

(8)

+ 1

M lLopp tPl mMl rmRmsSc

trm

vtsrmp(vrm) +dtrm atsrmp(arm)

+dtrm atsrmp(arm)

Recourse 2

Subject to:

vtsrm atsrm

Ptrm l Ld, m Ml, rm Rm, t Tl, s S (9)

vtsrm+ atsrm

Ptrm l Ld, m Ml, rm Rm, t Tl, s S (10)

atsrm , atsrm

0 l Ld, m Ml, rm Rm, t Tl, s S (11)

where:

vtsrm is the value of property rm in time tand simulation s.

E

ctrm

vtrmp(vrm ) = ctrm

M

sS

vtsrm

p(vrm ) defines the expected values of the properties rm whensent to the initial destinations.

atsrm andatsrm

are deviations from the targets Ptrm andPtrm

for propertyrmin periodt and simulations.

ctrm , dtrm

, dtrm , p (vrm), p (arm) and p

arm

remain unchanged from their original definition in Sec-

tion 2.2.1.

Recall that the first-stage properties vtrm are (non)-linear functions of the first-stage binary decisionvariables, z tlinit

cluslmk

; the first set of recourse deviation variables penalize the random events at the first-

stage destinations. The second set of recourse (second-stage) properties are (non)-linear functions of the

continuous processing variables yts

d(mld). If the optimization problem in Eq. (8)(11) were to be solvedfor each geological scenario individually, where the scenarios denote the combinations of simulations fromthe various sources (mines), it is possible to average to obtain the wait-and-see (WS) solution (Birge andLouveaux, 1997). Coupled with a solution from the previous optimization problem, it is possible to obtainthe expected value of perfect information (EVPI), the value one would pay to know the geology with certaintyin order to optimize their mining supply chain.

In a simplistic case where the supply chain consists of a mill and a waste dump, the second set ofrecourse variables would drop out, and Eq. (8) could be written similar to the two-stage stochastic integerprogram (SIP) proposed by Ramazan and Dimitrakopoulos (2012), with the exception of the mining capacityconstraints, which are assumed to be constant given the fixed production schedule and therefore do notinfluence the optimization of the supply chain. This formulation and model, of course, does not solve


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production scheduling, but rather the optimal destination policies for material coming from the mines andthe subsequent processing streams.

The proposed method for clustering and optimizing the mining supply chain under geological uncertaintyalso has many practical implications for operational mines. Recall that a solution to the problem outlined in

Eq. (8) to (11) generates destination decisions for each cluster, where the clusters have been defined usingall simulations for a mine and material. Consider the (almost certain) case where there is not a geologicalsimulation that represents exactly what is in the ground. For any block extracted, it is possible to calculateits cluster membership by finding the cluster that is closest in terms of block properties. This block shouldthen be sent to the destination that the cluster is sent to in the stochastic optimization problem solution. Inpractical terms, this provides an opportunity for very complex policies for operating mines, somewhat akinto cut-off grades, values or revenues. Since these initial cluster destination decisions have been made forpotentially complex mining supply chains, these destination policies are more useful in that they address thecore issue that cut-off policies cannot: blending operations do not consider the value of an individual block,a key assumption required for the application of cut-off grades or values.

2.2.5 Mining supply chain optimization using particle swarm optimization

Given that mixed integer non-linear programs can be extremely complicated to solve for problems of thismagnitude using mathematical programming, an algorithm using particle swarm optimization (PSO) withlocal heuristics is used. PSO is a population-based metaheuristic that uses cognitive and social behaviourbetween simulated particles to achieve a high quality (but not necessarily optimal) solution in a reasonableamount of time (Kennedy and Eberhart, 1995). Unlike many other metaheuristics, the proposed solutionscheme is suited for PSO because the method is capable of handling mixed integer non-linear problemswith ease. The goal of PSO is to have all particles locate optima in a multi-dimensional space, where eachparticle is initially assigned random positions and velocities. At each position update, the objective functionis sampled (i.e. the supply chain is evaluated), and the swarms best-valued solutions are given more weight.Through exploration and exploitation of known high-quality solutions, the particle cluster converges togetheraround an optimum or several optima (Brownlee, 2011).

Consider a set ofNp particles, where each particle is represented by its own distinct solution vector S ol,Sol or S ol (depending on whether the problem is for block destinations, deterministic cluster destinationsor stochastic cluster destinations, respectively). The current solution vector, or position vector, for a particle

p will herein be denoted by xp. Additionally, consider three vectors that are of the same size as the solutionvector: velp, a particle velocity vector; xp,best, which stores particle ps best solution found to date; andxglobal, a global best solution vector that stores the positions of the best solution found throughout thealgorithm. At iterationt + 1, a particles velocity is updated by using:

vp (t+ 1) =c1vp (t) +

c2 rand()

xp,best xp

+

c3 rand()

xglobal xp

(12)

wherevp (t+ 1) is the new velocity for the pth particle,c1,c2andc3are weighting coefficients for the particlescurrent inertia, its personal best solution and the global best solution, respectively, and rand() [0, 1] is arandom uniform random number. The particles velocity is therefore a combination of its own inertia at the

previous iteration, and an attraction towards its best solution and the global best solution. The particlesposition is then updated at iteration t + 1 using:

xp (t+ 1) =xp (t) +vp (t+ 1) (13)

where if any element xpj of the particle exceeds its bounds (defined in Section 2.2.2), xpj will be set to the

closest (minimum or maximum) bound.

After the velocity and position updates are performed for each particle, the solution is then evaluatedby using the methodology for supply chain evaluation presented in Section 2.1.3. Let f(xp (t)), f

xp,best

and f

xglobal

denote the current, best and global best objective function values, respectively, from Eq. (2)

at iteration t. If f(xp (t)) f

xp,best

, update the particles best solution vector to xp,best = xp (t). If f(xp (t)) f

xglobal

, update the global best solution vector to xglobal =xp (t).


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2.2.6 Localized improvements using heuristics

Several types of heuristics can be employed to quickly improve the quality of a solution and ensure thatthe particle is not trapped in a local minimum. Rather than applying heuristics on each of the particles(which can be time consuming) consider the global best particle solution vector xglobal (t) found to date at

iteration t of the PSO algorithm. Experimentally, it has been seen that it is often useful to employ one ofthe four heuristics:

1. When the particles have converged, freeze the block or cluster destinations,zd

blmk

orztd

cluslmk

and

perform the next set of iterations solely changing the processing stream decisions,y tsd(mld).

2. Restart each of the particles from the global best solution,xglobal (t), with new, random velocity vectors.

3. Randomly select a block or cluster destination decision variable, zd

blmk

or ztd

cluslmk

and change

it to a different candidate destination. Accept the new solution if it yields an improvement in theobjective function value, otherwise reject.

4. Randomly select a destination decision ytsd(mld) from the global best particle solution vector andrandomly choose whether to multiply or divide the value by 2. The related processing decision variablesneed to be re-normalized to guarantee material flowing from the destination does not exceed the amountflowing in. Accept the new solution if it yields an improvement in the objective function value, otherwisereject.

3 Application at Vales Onca Puma Nickel Laterite Mine

The proposed methodology is tested at Vales Onca Puma nickel laterite deposit. While the model used forthis case study does not demonstrate a full supply chain from pit to port, the blending streams are complexand can be used to highlight the features of the proposed method. For confidentiality purposes, most ofthe economic values and production capacities have been modified from what the mine uses in reality or arewithheld.

3.1 Overview of the operation

The Onca Puma nickel laterite mining complex is located 20 km north of Ouril andia do Norte, in ParaState, Brazil, and primarily uses a side-hill mining method. The Onca and Puma deposits are situatedapproximately 16 km apart, with an overall concession of approximately 40 km 2. The primary chemicalconstituents of interest within the nickel laterite profile are nickel, cobalt, silica, magnesia, alumina and iron.A strict control must be placed on the silica (SiO 2) to magnesia (MgO) ratio and iron (Fe) content to ensurethat the plant operates with minimal disruptions. A failure to deliver the appropriate plant feed blend canhave significant economical and operational implications.

The lithologies of the ore deposits can be simplified into three distinct layers: limonite, which has beencompletely altered by chemical weathering and contains some nickel, low MgO and high Fe; saprolite, whichis the primary source of nickel for the ferronickel pocess and contains of low Fe and high MgO; and bedrock,

which consists of low nickel grades, high MgO and low Fe. For the purpose of this study, only the On ca depositwill be considered. Within the deposit, five material types are considered and are classified as: bedrock,limonite, saprolite waste (less than a predetermined nickel waste cut-off), low-grade saprolite (between thenickel waste cut-off grade and intermediate cut-off grade) and high-grade saprolite (above the intermediatenickel cut-off grade). While the mine uses a stockpile classification scheme based on the nickel grade, silica tomagnesia ratio and material type, the previously outlined classification scheme based solely on nickel gradeis used in order to let the optimizer a high degree of flexibility in decision-making.

Figure 1 shows a diagram for material flow at the mine used for this study. The alternative stockpilescheme has seven initial destinations: three low-grade stockpiles, three-high grade stockpiles and a wastedump. The low- and high-grade stockpiles are short-term stockpiles with a total capacity ofTSPmaxdry tonnes,where the capacities are all equal in size but the tonnage is withheld. For this study, material is permitted to


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carry-over between periods in these intermediate stockpiles. Each of the six intermediate stockpiles then feedtwo homogenization piles, each having a capacity ofTHmax dry tonnes per period. The homogenization pilesalternate between feeding the pyrometallurgy processing plant and being filled by the intermediate stockpilesevery 36 days (considered as one production period in this study); during the filling period, material cannotflow from the homogenization pile to the processing plant. The processing plant has a THmaxdry tonne capacity,

and has strict requirements on the blended feed from the homogenization piles. The silica-to-magnesia ratiois of primary importance and must lie between 1.5 and 1.8. Additionally, the plant has a requirement thatthe iron grade (Fe) lies between 12% and 16%. Because of these strict constraints on the blended feed tothe processing plant, conventional mine planning optimizers are often inapplicable and generate unrealisticsolutions due to the assumption that the value can be calculated based on the blocks nickel grade; in thecase of a nickel laterite mining operation, the nickel grade is of secondary importance to the blended plantfeed specifications.

Figure 1: Schematic of material flow at the Onca Puma nickel laterite mining operation.

A set of 10 simulations of the lateritic profile was simulated using the single normal equation simulation(SNESIM) algorithm (Strebelle, 2002) for the Onca deposit; in particular, the limonite, saprolite and bedrocklayers were simulated to quantify the high variability and volumetric uncertainty for the deposit. For eachlithological simulation, the saprolite layer is retained for further joint simulation of nickel, magnesia, silica,iron and the dry tonnage factor using the direct block Min/Max Autocorrelation Factor simulation method(Boucher and Dimitrakopoulos, 2009; Goodfellow et al., 2012). The simulations show substantial variabilityfrom one simulation to the next, and, as will be seen in the proceeding sections, is quite challenging for theproposed method to find an adequate solution. In total, 20 simulations are used for the Onca deposit on 12.5 12.5 3 m3 blocks in the x-, y- and z-directions, respectively. It is noted that the mine generally doesshort-term production scheduling on 6.25 6.25 3 m3 block sizes, however the larger block sizes were kept

for comparative purposes with the estimated (kriged) models supplied by the mine.

The mine production schedule is based on 36-day periods (the time to fill a homogenization pile). Theproduction schedule is generated from a portion of the mines annual production schedule by sequentiallymining each bench until the short-term total mining capacity is met (950,000 dry tonnes). It is notedthat because this study does not consider the Puma deposit, the schedule has been condensed to representcontinuous mining at Onca. The production schedule therefore spans over 44 consecutive periods and contains69,877 blocks.


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3.2 Proposed models

The Onca Puma mining complex (Figure 1) is modelled using the methodology proposed in Section 2.1.In specific, the block, nickel, silica, iron and magnesia tonnages are properties that are passed through thesupply chain from the mine through the stockpiles and homogenization piles, and finally to the plant. It is

assumed that the intermediate low- and high-grade stockpiles are able to carry over material from one periodto the next for strategic purposes, with a lag time of one period. The homogenization piles are not permittedto carry over material between production periods, thus all material that goes to either of these piles is sentto the plant in the subsequent period. At no point does the algorithm consider the economic value of anindividual block; the economic value of the processed metal, the silica-to-magnesia ratios and the iron gradeare calculated at the plant for the blended material coming from the homogenization piles.

3.2.1 Deterministic model

The optimization model for the optimization of deterministic orebody models (not considering geologicaluncertainty) is as follows.

Objective:

max44t=1

vtMV

(1.008)t

6s=1

dtSCs

atSCs

dtMC

atMC

dtMC

atMC

44t=1

dtMSiMg

atMSiMg

+dtMSiMg

atMSiMg

+dt

MF e%

atMF e%

+dtMF e%

atMF e%

(14)

where:

vtMNi,vtMSiO2

,vtMMgO, vtMFeand v

tMT are the nickel, silica, magnesia, iron and total tonnages sent to the

processing plant in periodt. Recall from Section 2.1.3 that these values are calculated by algorithmicallyevaluating the supply chain by sending the blocks (or clusters) from the mine through the blending orprocessing paths.

vtMV is the economic value of the nickel recovered from the blended feed of material at the processingplant in periodt. Using a nickel price, pNi , a recovery, rNi and a processing cost,cproc, all of which arenot disclosed for confidentiality purposes, the economic value of the recovered nickel can be calculatedas follows:

vtMV = vtMNi pNi rNi v

tMT cproc (15)

atMC and atMC are deviations from the upper and lower plant capacity targets, and are calculated as

follows:

atMC= max

0, vtMT THmax

(16)

atMC= max

0, THmax vtMT

(17)

at

MSiMg and at

MSiMg are deviations from the upper and lower silica-to-magnesia ratio constraints,respectively, and are calculated using the following non-linear constraints for the blended feed at theplant:

atMSiMg= max

0,

vtMSiO2vtMMgO

1.8

(18)

atMSiMg= max

0, 1.5

vtMSiO2vtMMgO

(19)

atMF e% anda

tMF e% are deviations from the upper and lower iron grade constraints, and are calculated

using the following non-linear constraints:


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atMF e% = max

0,

vtMF evtMT

16

(20)

atMF e% = max

0, 12

v tMF evtMT

(21)

vtSTs is the value of the stockpile tonnage for stockpile sand period t. atSCs is the deviation from the total capacity for stockpile s {1, . . . , 6} in period t and is calculated

as follows:atSCs = max

0, vtSTs T

SPmax

(22)

The penalty costs associated with deviations from the silica-to-magnesia ratio, dtMSiMg and dtMSiMg,

are both set to 30,000,000 to heavily penalize deviations that could disrupt the plant productivity.

The penalty costs associated with stockpile capacity deviations, dtSCs , are set to 1 for the first twolow-grade stockpiles, 1.125 for the third low-grade stockpile, 1.5 for the first two high-grade stockpilesand finally 1.625 for the third high-grade stockpile. It was found through experimentation that thePSO algorithm tends to put more material into the last stockpile and has difficulties moving backwardsto lower stockpiles. The higher value helps the heuristics move material back to the other stockpiles

from the final stockpile of the set. The penalty costs associated with plant capacity deviations, dtMC and d

tMC, are set to 2.

The penalty costs associated with deviations from the upper and lower iron grade targets, dtMF e% and

dtMF e%, are set to 20,000,000 to heavily penalize deviations that could disrupt the plant productivity.

3.2.2 Stochastic model

The stochastic model for the Onca deposit is similar to the deterministic model, however it evaluates the effectthat each simulation has on the supply chain and penalizes deviations for each of the geological simulations,effectively leading to more robust destination decisions. The model is as follows:

Objective:

max44t=1

E{vtMV}

(1.008)t

20g=1

6s=1

dtSCs

at,gSCs

dtMC

at,gMC

dtMC

at,gMC

20g=1

44t=1

dtMSiMg

at,gMSiMg

+dtMSiMg

at,gMSiMg

+dt

MF e%

at,g

MF e%

+dtMF e%

at,g

MF e%

(23)

where:

1. g {1, . . . , 20}is used to denote the twenty geological simulations used to represent the On ca orebody.

2. The definition of all deviation variables and property values from the deterministic formulation havebeen modified by adding an additional superscript g to represent the value of the deviation or propertyvalue in simulation g .

3. E{vtMV} is used to denote the expected value of the recovered nickel metal of the blended material at

the plant, i.e. E{vtMV}= 120

20g=1

vt,gMV andv

t,gMV is the value of the recovered nickel metal at the plant in

time t and simulation g .

4. All penalty costs remain the same as the deterministic model.

3.3 Numerical Results

3.3.1 Deterministic optimization

The model and mathematical formulation presented in the previous section are tested on a deterministic(kriged) orebody model given by the mine. Given that mine supply chain optimization is a new area of


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research, an analysis of the performance of the proposed methodology on a deterministic orebody modeldemonstrates the methods effectiveness under more simplistic and conventional circumstances.

Two cases using deterministic orebody models are tested: the first with block destinations, discussed inSection 2.1.3, and the second using clusters, discussed in Section 2.2.3. Theoretically, the solution generated

by solving block destinations should outperform the clusters, given that the optimizer has more control overthe decision-making, however, as will be demonstrated, this control comes with a substantial amount ofadditional computing time; it is often more practical to generate high-quality solutions quickly. Recall thatin the case of block destinations, there are 69,877 block destination decisions (i.e. the number of blocks inthe model); in the case of clusters, the number of decision variables is reduced to nclusters 44, where 44 isthe number of production periods considered. The block destination heuristic (method 3 in Section 2.2.6) isemployed far less for the block destination problem because of the amount of time required to re-evaluatethe problem; performing the heuristic on the block destination problem takes approximately 90 minutes totest 10% of the blocks, whereas the cluster destination takes approximately 5 minutes, and can therefore bedone much more frequently.

Both problems are solved using 20 particles, and in the case of the clustering problem, 50 clusters eachare used for low-grade and high-grade saprolite. This results in reducing the block destination problem

from 69,788 decision variables to 4,532 variables (waste, limonite and saprolite waste are each assigned to adifferent cluster). Through trial and error, the particles inertia is set to 0.6, the particles best inertia is setto 0.25 and the global best inertia is set to 0.8 (Eq. 12). The block destination problem ran for 9 hours and55 minutes and performed 750 iterations, whereas the cluster problem ran for 7 hours and 43 minutes andperformed 1250 iterations.

Figure 2 shows they key indicators of the performance of the algorithm. It can be seen that the problemsolved using clusters always meets the minimum silica-to-magnesia ratio (1.5) and never exceeds the upperlimit of 1.8. The solution using blocks, however, only meets the minimum ratio requirement on a fewoccasions, meaning that the solution is infeasible and would likely shut down the plant if used in practice.The plant feed capacities (hence homogenization pile tonnages) are never exceeded in the clustering solutionand on some occasions are not able to meet the targets. This solution is substantially better than the blocksolution, which often drastically exceeds the plant feeds capacity. In both cases, the minimum and maximum

bounds on the plant feeds iron grade is always satisfied. Figure 2 also shows the cumulative net presentvalue (NPV) of the solutions, expressed as a percentage of the final cumulative NPV for the deterministicblock problem; this convention will be adopted herein to express economic values. While the NPV for theblock destination solution may be quite a bit higher than the cluster solution, the fact that the solution farexceeds the silica-to-magnesia ratios implies that the solution is not realistic. Figure 3 shows a comparisonbetween the deterministic block and cluster destination problems for the tonnages at each of the six stockpilesover time. In general, it can be seen that the low-grade stockpiles are often not used to capacity (with theexception of the third stockpile). The high-grade stockpiles, however, are much more used and the capacitiesare often exceeded. Given the under-utilization of the low-grade stockpiles, it is assumed that some of thepeaks in stockpile tonnages can be smoothed out by temporarily placing the material in other stockpiles,which could be confirmed by re-optimizing the problem, but is not considered in this case study.

It should be once again mentioned that the problems should generate similar solutions, however, theamount of time required to solve the block destination problem is impractical for this case study. A large partof the reason why the block destination solution for the deterministic case is not obeying the key indicatorsshown in Figure 2 is due to the size of the solution vector. The particle swarm optimization algorithm has beendocumented to perform less effectively in cases with high-dimension solution vectors (Li et al., 2011; Yanbin,2008). As a result of the higher dimension (more block decision variables), the greedy block destinationheuristic used is less effective because it takes substantially longer to run. For reasons of practicality interms of time and effectiveness, it can therefore be stated that solving the problem using cluster decisionsrather than block decisions is much more appealing. Future work will focus on disaggregating, optimizingand re-aggregating the clusters to improve the quality of the solution and reduce some of the instabilitiesassociated with the k-means clustering algorithm; however, as shown in Figure 2 and Figure 3, the clustersand optimizer do indeed perform extremely well for the deterministic case.


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Figure 2: Graphs comparing the performance of the PSO algorithm when using block destinations or clusterdestinations.

Figure 4 shows an analysis of the effect that the number of particles has on the average objective functionvalue (Eq. 2); note that the objective function value itself is not disclosed, however the values are expressedrelative to the first iteration with 5 particles. It can be seen that as more particles are used, the solutions tendto converge on an optimum in fewer iterations, however the final objective function values remain relativelyunaffected, regardless of the number of particles used (within 4%). The solution times for the deterministicproblem with clustering using 5, 10, 15 and 20 particles were 288, 302, 363 and 549 minutes, respectively.The tests for 5, 10 and 15 particles are close in time because the running time of the algorithm is dominatedby the heuristics used. Due to the faster convergence (and despite the larger algorithm running time), alltests are performed using 20 particles, unless otherwise stated.

Figure 5 shows an analysis of the number of clusters used, and was generated using only 10 particles.Naturally, one would expect that as the resolution of the decisions decreases (i.e. aggregating more blockinformation into a single cluster), the quality of the resultant solution would rapidly deteriorate. It can beseen while this is true, creating too many clusters (e.g. 100) is excessive and does not yield a substantiallybetter solution; recall that as more clusters are added, the dimension of the optimization problem alsoincreases. The higher dimensions also comes at a cost of longer solution times, which were 113, 120, 122, 140,302 and 2158 minutes for 5, 10, 15, 25, 50 and 100 clusters for the low- and high-grade saprolite material,respectively. It is noted that the solution generally does converge faster as more clusters are used, but thedifference in value between 25, 50 and 100 clusters is not substantial, and therefore does not indicate anyadded value for the associated computation time. Figure 5 also demonstrates some of the instabilities that canoccur when choosing different starting points for the k-means clustering algorithm; in particular, it is noted


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Figure 3: Comparison of tonnages going to the low-grade (LG) and high-grade (HG) stockpiles.

that the solution using 5 clusters is approximately 1.5% higher than the solution using 10 clusters. Whilethese differences are essentially negligible, future work will investigate disaggregation and re-aggregationapproaches during the algorithms runtime to improve the stability of the clusters. For the purpose of thisstudy, only 50 clusters for both the low- and high-grade saprolite will be used, as it balances solution qualityand computation time.


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Figure 4: Analysis of the influence on number of particles on the objective function over the course of the

algorithm.

Figure 5: Analysis on the quality of the solution with respect to the number of clusters used.

3.3.2 Risk Analysis of deterministic destination policies

As mentioned in Section 2.2.4, moving from block destination decisions to cluster destination decisions permitsthe optimization of the supply chain (in this case, blending streams) under geological uncertainty. By

comparing the location of the clusters (metal content, tonnages, etc.) to each block in each simulation, itis possible to obtain the cluster membership of the block in each simulation. As a result, each cluster mayhave varying values for the properties (i.e. nickel, silica, magnesia, iron and total tonnages) across differentsimulations, but the decision-making can be done on a cluster-basis, hence tie decision making together acrossall simulations. Consider the previous deterministic case for the estimated model. Each cluster contains a setof coordinates for nickel, iron, silica, magnesia and tonnage and an assigned destination from optimization.For any given simulation and block, the cluster that closest matches the blocks coordinates will be sent tothe clusters assigned destination. The clusters then become a more advanced policy for material destinationsand can take into account much more information than cut-off grades or values and are more suitable forblending operations.


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To highlight the risk in the cluster destination decisions from the deterministic case study from theprevious section, the original cluster locations and destinations are kept. The PSO algorithm is used onlyto optimize the processing stream options across the various simulations, however the cluster destinationdecisions remain static for all simulations. Figure 6 shows a risk analysis for the key parameters for thecluster destinations obtained from solving the deterministic problem with kriged orebody model. It can be

seen that when using the destination policies from the conventional model, there is an extremely low chancethat the minimum silica-to-magnesia ratios will be met, which would likely stop the plant entirely. This is aresult in the difference between the distributions between a kriged model and a simulated model; it is knownthat the distributions for simulations can be quite different than the estimated model; hence, the clusterslocations for the estimated model do not appear to be adequate for the simulations. The plant tonnagealso appears to be extremely variable, however this is a common occurrence when analyzing solutions withsimulations for this deposit and is largely based on the local distribution of lithologies within the simulation;it is possible that these fluctuations can be reduced through stochastic production scheduling. For the mostpart, the iron grades obey the minimum and maximum constraints. The cumulative NPV for the simulations,however, are substantially different from the deterministic model using clusters in Figure 2 and show a net loss(note that the NPV has been scaled by the maximum cumulative NPV from the deterministic model usingblock decisions). Again, these substantial differences are attributed to the large differences in distributions

between the simulations and the estimated model.

Figure 6: Risk analysis for the cluster decisions made for the deterministic orebody model.


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3.3.3 Stochastic optimization

Similar to the deterministic model, each of the simulations can be optimized independently from the othersto give an upper bound on the optimal solution using the deterministic model presented in the previoussection; in stochastic optimization literature, this is referred to as the wait-and-see (WS) solution (Birge

and Louveaux, 1997). This is similar in concept to mine planning where each simulation can be used togenerate an independent production schedule to obtain the Expected Value of Perfect Information (EVPI),which can be useful to understand how much value there is in knowing the geology deposit perfectly. Thisis particularly important for the proposed methodology because, given that it involves metaheuristics andnon-linear optimization, a theoretical bound on the solution is not easily obtained. Figure 7 shows eachsimulations profiles when solved independently (including independent clusters). Unlike the results shownin Figure 6 (risk in the deterministic cluster solution), the results indicate that it is indeed possible to meetthe limits silica-to-magnesia ratios, with only a few deviations. Additionally, it can be seen in Figure 7 thatthe plant feed capacities are rarely exceeded, despite the fact that there is substantial risk in being ableto supply enough material; again, this is related to the variability of the lithologies of the simulations andis not related to the algorithms abilities. The minimum and maximum bounds on the iron grade are alsoobeyed during the life of the schedule. Figure 7 also indicates that it is possible to obtain a positive net

present value for the given production schedule if the geology is known perfectly. This highlights the fact thatthe clusters and optimal clustering destination decisions made for the deterministic model are inadequate

Figure 7: Risk analysis for the cluster decisions when each simulations destinations are optimized individually(perfect knowledge).


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when considering uncertainty, and that it may be possible to generate a more robust destination policy thatconsiders geological uncertainty.

Using the stochastic optimization model outlined in Section 3.2.2, it is possible to link the geological sim-ulations together in the optimization process through clusters and the cluster destination decision variables,

which can in turn be used as a robust destination policy, somewhat akin to an advanced cut-off grade policyunder geological uncertainty. The solution time for optimizing with this method is substantially longer forthe stochastic case than the deterministic case; to solve the formulation with 50 clusters for the low- andhigh-grade material and 20 particles, it takes over 36 hours to perform 1250 iterations, where there is nosubstantial change in objective function value. The reason for this drastic increase is that the optimizer needsto evaluate a supply chain for each simulation and each particle for every iteration of the algorithm. Whilethe algorithm is already parallelized on the computers CPUs, it may be possible to drastically increase theperformance by evaluating the supply chain on a GPU, which is a topic of future research. It is also possibleto increase the performance by decreasing the number of particles or clusters, however these have been keptconsistent for benchmarking purposes.

Figure 8 shows the risk profiles for the stochastic destination solution. It can be seen that it is possible tomake robust cluster destination decisions that meet both the plant feeds silica-to-magnesia ratio requirements

and the plants iron grade requirements with very minor deviations. When compared to the previous case inFigure 7, there are actually fewer deviations from the silica-to-magnesia ratio and iron grade. This is caused

Figure 8: Risk analysis for the stochastic destination optimization solution where a common destinationpolicy is applied to all simulations.


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by the fact that the formulations are slightly different: the stochastic model considers the expected valueof the blocks but continues to penalize each of the constraint deviations heavily whereas the previous casehas more of a balance between net present value and deviations. The plant feed tonnages are often lowerthan the target ofTHmax tonnes, however the performance is not worse than the WS solution; the plant feedcapacities are never exceeded, which indicates a high level of spatial variability in the saprolite layer between

the simulations. The solution does generate a positive net present value for the majority of the simulations.The lower net present value for the stochastic solution when compared to the WS case is also normal giventhat the destination policies have been designed to be robust for the simulations, whereas the previous casewill only generate solutions that are optimal for that specific simulation. While there appears to be a 20%chance that the schedule and destination policy will generate a negative net present value (loss), one mustconsider the high degree of variability between the simulations making it particularly challenging to generatea robust destination policy, and that these simulations also indicate a very low WS solution cumulative NPVin Figure 7.

4 Conclusions

This paper proposes a mining supply chain modelling methodology and is optimized using a particle swarm

optimization algorithm with local heuristics. Given that the number of decision variables to direct materialfrom a block to its initial destination can be orders of magnitude more than decision variables required tomodel the remainder of the supply chain, and that the bulk of the algorithms time is spent evaluatingthese initial destination decisions, a method for aggregating blocks using the k-means clustering algorithmis proposed. These aggregations have been shown to substantially reduce the size of the problem, thus thecomputation time of the algorithm, with minimal loss in solution quality due to aggregation. It is possibleto extend the concept of the aggregation into the context of supply chain optimization under geologicaluncertainty, whereby the aggregates are created based on the set of all simulations at once. This optimizationproblem can then be reformulated into a mixed integer nonlinear stochastic program, where the first stagedecisions choose the optimal destinations for the clusters and the recourse decisions optimize the remainderof the supply chain. Using this method, the aggregates and optimal aggregate destinations can then be usedas a destination policy that is robust under geological uncertainty.

The proposed method is tested at Vales Onca Puma mine in Para State, Brazil. Experimental resultsindicate that the proposed method works well for optimizing with complex blending operations with non-linear constraints. Additionally, the proposed method of stochastic optimization for the destination problemperforms well and generates a single robust destination policy that adheres to the strict blending requirementsat the processing plant. Future work will seek to improve cluster stability by disaggregating and re-aggregatingthe clusters as the algorithm progresses, test the proposed methodology on other mining supply chains andintegrate production scheduling with the supply chain optimization under uncertainty in the context ofstochastic global optimization.

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Birge, J.R., Louveaux, F. (1997) Introduction to Stochastic Programming. Springer Series in Operations Research,Berlin.

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mining supply chain optimization under geological uncertainty

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